Hamiltonian Truncation Effective Theory

Abstract

Hamiltonian truncation is a non-perturbative numerical method for calculating observables of a quantum field theory. The starting point for this method is to truncate the interacting Hamiltonian to a finite-dimensional space of states spanned by the eigenvectors of the free Hamiltonian H_0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> with eigenvalues below some energy cutoff E_\text{max} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>E</mml:mi> <mml:mtext mathvariant="normal">max</mml:mtext> </mml:msub> </mml:math> . In this work, we show how to treat Hamiltonian truncation systematically using effective field theory methodology. We define the finite-dimensional effective Hamiltonian by integrating out the states above E_\text{max} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>E</mml:mi> <mml:mtext mathvariant="normal">max</mml:mtext> </mml:msub> </mml:math> . The effective Hamiltonian can be computed by matching a transition amplitude to the full theory, and gives corrections order by order as an expansion in powers of 1/E_\text{max} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:msub> <mml:mi>E</mml:mi> <mml:mtext mathvariant="normal">max</mml:mtext> </mml:msub> </mml:mrow> </mml:math> . The effective Hamiltonian is non-local, with the non-locality controlled in an expansion in powers of H_0/E_\text{max} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>/</mml:mi> <mml:msub> <mml:mi>E</mml:mi> <mml:mtext mathvariant="normal">max</mml:mtext> </mml:msub> </mml:mrow> </mml:math> . The effective Hamiltonian is also non-Hermitian, and we discuss whether this is a necessary feature or an artifact of our definition. We apply our formalism to 2D \lambda\phi^4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> theory, and compute the the leading 1/E_\text{max}^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mtext mathvariant="normal">max</mml:mtext> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> corrections to the effective Hamiltonian. We show that these corrections nontrivially satisfy the crucial property of separation of scales. Numerical diagonalization of the effective Hamiltonian gives residual errors of order 1/E_\text{max}^3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mtext mathvariant="normal">max</mml:mtext> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> , as expected by our power counting. We also present the power counting for 3D \lambda \phi^4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> theory and perform calculations that demonstrate the separation of scales in this theory.

Topics & Concepts